An object is thrown vertically from a height of 5 "m" at 3 "m/s". How long will it take for the object to hit the ground?

1 Answer
Mar 4, 2016

0.7 quad "s"

Explanation:

"Thrown vertically" could mean either upwards or downwards. I am assuming it to be downwards.

Step 1: Understand the problem.

An object is under free fall. It is accelerating at g = 9.81 quad "m/s"^{2} downwards. It is given an initial velocity of 3 quad "m/s" downwards.

Step 2: Identify the relevant equations.

To find displacement, s, under constant acceleration, a, and initial velocity, v_0, after time, t, the following equation is applicable:

s = v_0 t + 1/2 a t^2

To get the time as a function of the distance traveled, make t as the subject of formula. Begin by writing the quadratic equation in standard form.

t^2 + frac{2 v_0}{a}t - frac{2 s}{a} = 0

Subsequently, complete the square.

(t + frac{v_0}{a})^2 - frac{2 s}{a} = (frac{v_0}{a})^2

(t + frac{v_0}{a})^2 = frac{v_0^2}{a^2} + frac{2 s}{a}

= frac{v_0^2 + 2as}{a^2}

Reject the negative square root and take the positive one. This is because we only restrict ourselves to t > 0.

t + frac{v_0}{a} = frac{sqrt{v_0^2 + 2as}}{a}

t = frac{sqrt{v_0^2 + 2as} - v_0}{a}

Step 3: Set up the coordinate system.

I let the downwards direction be positive. The ground level is arbitrarily set to be the origin (i.e. s = 0). This is up to your preferences.

Step 4: Identiy the parameters.

In this question,

  • a = g = 9.81 quad "m/s"^2

  • v_0 = 3 quad "m/s"

  • s = 5 quad "m"

Step 5: Plug into the equation.

t = frac{sqrt{v_0^2 + 2as} - v_0}{a}

= frac{sqrt{(3 quad "m/s")^2 + 2(9.81 quad "m/s"^2)(5 quad "m")} - (3 quad "m/s")}{9.81 quad "m/s"^2}

= 0.749 quad "s"

Now earlier, I assume that the object has an initial velocity of 3 quad "m/s" downwards. The formula also works if the initial velocity is 3 quad "m/s" upwards. In that case, everything remains the same, except

  • v_0 = - 3 quad "m/s"

Plugging in the values again

t = frac{sqrt{v_0^2 + 2as} - v_0}{a}

= frac{sqrt{(-3 quad "m/s")^2 + 2(9.81 quad "m/s"^2)(5 quad "m")} - (-3 quad "m/s")}{9.81 quad "m/s"^2}

= 1.361 quad "s"